**Adding household debt and curvature to the IS curve yields surprising insights**

*Fed Research and why a new curve was built*

The Cubic IS curve is built upon a consumption function that was described in this research paper from the Federal Reserve Bank of Dallas (An IS-LM analysis of the zero lower bound, 2011). In essence, the paper laid out a simple IS/LM model, but restricted the IS curve to be a consumption function, rather than including other factors like government spending. The authors spelt out the traditional liquidity trap situation where, after a shock, traditional monetary policy has lost its traction. Diagrammatically, this is represented below:

In this situation, the monetary authorities could still influence the economy and achieve a new equilibrium via their effects on inflation expectations. These expectations were embedded in the IS/consumption function, so it the central bank was able to raise this sufficiently, it could move the IS curve upwards towards a more normal equilibrium.

This research led to some interesting questions: how could the build-up of debt in an economy be reflected in the IS curve and in a way that results in a bursting of the bubble? Moreover, how could this debt-enhanced consumption function reveal the competing influences in a liquidity trap scenario of unconventional monetary policy and deleveraging from private agents?

A method to address both of these areas was to allow debt to add curvature to the IS curve, which lent itself to become represented as a cubic function.

*The Cubic IS consumption function & its interpretation*

* *The consumption function from the Fed research was as follows:

Where R is the nominal interest rate, C refers to (logarithm of) real consumption at time t, and σ is the ‘pure rate of time preference’, Π is inflation, and ρ refers to the ‘elasticity of intertemporal substitution’.

This formed the basis of the cubic function, as total debt and the number of households (not) in the debt market are added to a cubic specification:

D refers to the total debt, α is the number of households without debt (not in the debt market), λ is defined as 1/σ and d =

Referring to a paper by RWD Nickalls titled ‘A new approach to solving the cubic: Cardan’s solution revealed’, we can gain an insight into the main parameters that determine the shape of the cubic function: The paper generated key parameters that shape the function, of which we discuss the main two here.

In the context of the present model, delta has the function:

The other main parameter is Rn:

It can be seen in the above that the main determinants of the shape of the cubic are D and alpha. To summarise the basic effects of changes in these variables, the effects are:

- A rise in D raises delta
- As alpha gets bigger, delta gets smaller.
- A rise in small d raises R
_{n}(which we use to model a rise in inflation expectations) - As alpha gets bigger R
_{n}gets bigger. - As D gets bigger R
_{n}gets bigger

These parameters and their relationships help us to understand the effects of our economic variables on the shape and location of the cubic function.

Using the basic interpretation of the IS, the trade off in this partial equilibrium is between savings on bonds and consumption on output. So this gives us the level of consumption for a firm level of interest rates. When the interest rate is high, the opportunity cost of consumption is high, so savings on bonds is preferable over consumption. As the interest rate falls the opportunity cost of consumption falls too, so consumption becomes more likely.

It is the presence of high levels of debt that distort this relationship. With a highly curved IS curve, the required fall in rates to induce a certain amount of consumption is larger than in a linear case with no household debt. One may reason this as something akin to households being debt constrained at high levels of rates.

*A heterogeneous household sector completes the consumption function*

The two additions to the consumption function are total debt, D, and the number of households (not) in the debt market, alpha. To ascertain how these are obtained in the model, let debt per household be determined by the following equation:

Where S=lending standards, or average loan to value ratio, or average collateral rate.

Therefore, total debt in the economy is equal to the area under the curve:

The conclusion from this diagram and associated equations is that as standards drop, or the average collateral rate increases, the line becomes flatter and as it is anchored at hp e it moves out to include more houses in the debt market. Similarly, if there is an increase in hp e it acts like a financial accelerator: it induces households already with debt to take on more, and also brings in new households. So this is a double effect in debt. This situation probably more closely describes Australia. Whereas the UK, US and Ireland had both an increase in hp e and a lowering in standards.[1]

This completes the basics of the model. We now have enough detail to explore how a debt bubble might burst and to examine the competing forces of deleveraging and inflation expectations at the zero lower bound. This will be discussed in the next post.

[1] Empirical and anecdotal evidence for this relationship between lending standards, expectations and household debt, see http://www.frbsf.org/publications/economics/letter/2010/el2010-01.html this was also addressed by Mark Thoma at www.economistview.typepad.com

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